This page on thermopolia reports a quotation from Mary Beard, classics professor at Cambridge University:

“The best way to escape a diet of bread, cheese and fruit, eaten in small lodginggs over a shop or workshop, where there were limited or no facilities for cooking anything more interesting, was to eat out. Pompeii has long been thought of as a cheap café culture, with bars, taverns and thermopolia (as they are often called in modern guidebooks, though this was certainly not the standard ancient term) lining the streets, catching the passing trade—from visitors with time on their hands to local residents with nowhere nice of their own to be. In fact the masonry counters facing the pavements, with large jars (dolia) set into them and display stands behind, are one of the most familiar elements of in the Pompeian street scene.”

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pŏpīna is the one, borrowed from Oscan or Umbrian, and cognate with (native Latin) coquina. Indeed, a Packhum search gives no results for thermopolium and 54 results (59 matches) for popina - note that, still, thermopolium can be found e.g. in Plautus and Petronius.

Note that caupona and taberna were also common, but while L&S lists them as synonyms, they meant something slightly different. The former was more precisely a place where you could also sleep. The latter term, if not qualified with a following adjective (libraria, unguentaria, etc.), indicated a place where only or mostly wine was offered.

Here is an exhaustive excerpt from Gustav Hermansen's "Ostia: Aspects of Roman City Life" (he gives credit to Tönnes Kleberg in the first line):

The main result of his research is that there are six important names for establishments in this category: caupona, deversorium, hospitium, popina, stabulum, and taberna. These six words are distributed three ways:

The caupona, deversorium, hospitium, popina and stabulum denote what modern languages would call hotels: houses where travellers could put up for the night and at the same time could have their meals and could drink. A stabulum originally meant a stable, and in this connection it stands for a hotel where one could stop in with one's horse. As Latin literature shows, however, it did not matter whether one brought a horse or not!

A caupona was a hotel which catered to all of a traveller's needs, but eventually the value of the word was down-graded and acquired the connotation of a low-grade saloon, then was dropped from the language altogether.

A popina was a restaurant which could serve food and drinks to a customer but which did not offer any accomodation.

A taberna was comprehensive, meaning a shop in which all kinds of trades could be plied. To differentiate, the word was qualified with adjectives like (taberna) libraria, unguentaria, vinaria, or, [as seen above], cauponia. Eventually the meaning of a drinking place (tavern) prevailed when the word was without an adjective. A taberna would normally offer wine, rarely food.